In this work, we generalize the mass-conserving mixed stress (MCS) finite element method for Stokes equations [Gopalakrishnan J., Lederer P., and Sch\"oberl J., A mass conserving mixed stress formulation for the Stokes equations, IMA Journal of Numerical Analysis 40(3), 1838-1874 (2019)], involving normal velocity and tangential-normal stress continuous fields, to incompressible finite elasticity. By means of the three-field Hu-Washizu principle, introducing the displacement gradient and 1st Piola-Kirchhoff stress tensor as additional fields, we circumvent the inversion of the constitutive law. We lift the arising distributional derivatives of the displacement gradient to a regular auxiliary displacement gradient field. Static condensation can be applied at the element level, providing a global pure displacement problem to be solved. We present a stabilization motivated by Hybrid Discontinuous Galerkin methods. A solving algorithm is discussed, which asserts the solvability of the arising linearized subproblems for problems with physically positive eigenvalues. The excellent performance of the proposed method is corroborated by several numerical experiments.

翻译：本文工作中，我们将斯托克斯方程的质量守恒混合应力（MCS）有限元方法[Gopalakrishnan J., Lederer P. 与 Sch\"oberl J., A mass conserving mixed stress formulation for the Stokes equations, IMA Journal of Numerical Analysis 40(3), 1838-1874 (2019)]——该方法涉及法向速度与切向-法向应力连续场——推广至不可压缩有限弹性力学问题。通过引入三场Hu-Washizu变分原理，将位移梯度与第一类Piola-Kirchhoff应力张量作为附加场引入，从而规避了本构关系的求逆运算。我们将位移梯度出现的分布导数提升至一个正则的辅助位移梯度场。在单元层级可实施静力凝聚，最终形成待求解的全局纯位移问题。我们提出一种受混合间断伽辽金方法启发的稳定化方案。文中讨论的求解算法确保了在物理特征值为正的问题中，所衍生的线性化子问题的可解性。若干数值实验验证了所提方法的优异性能。